Some properties of convolutions of Pascal and Erlang random variables

This article studies the convolutions of Pascal random variables, i.e., negative binomial distributions with integer parameters. We show that the probability distributions of such convolutions can be expressed as a generalized mixture (i.e., mixture with negative and positive coefficients) of a finite number of Pascal distribution functions. Based on this result, further study on the limiting behavior of the failure rate function of the convolution is presented. A sufficient condition is given to establish the likelihood ratio order between two convolutions of Pascal random variables. Similar results are obtained for the convolutions of Erlang random variables.